Springer basic sets and modular Springer correspondence for classical types |
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Institution: | School of mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China;School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
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Abstract: | We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic , and with coefficients in a field of odd characteristic ): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols. |
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Keywords: | Modular representation theory Springer correspondence Reductive algebraic groups Classical types |
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